Everything that is True is Known to be True

(i) Assume that If “proposition p is true” then it is possible that it is known that “proposition p is true”. p →◊Kp
(ii) Assume that Both “proposition p is true” & it is not known that “proposition p is true”. p & ~Kp

(iii) Assume that it is known that “proposition p is true” if and only if “proposition p is true”. Kp ≡ p
(iv) Assume it is known that both “proposition p is true” & “proposition q is true” if and only if it is known that “proposition p is true” & it is known that “proposition q is true”. K(p & q) ≡ Kp & Kq

*Substitute the p in (i) with (ii), and we get [(p & ~Kp) → ◊Kp]. This means (i*) If both “proposition p is true” & it is not known that “proposition p is true”, then it is possible that it is known that “proposition p is true”.

(v) It is possible that it is known that both “proposition p is true” & it is not known that “proposition p is true”. ◊K(p & ~Kp) [Logical Consequence of (i*) & (ii) by Modus Ponens]

(vi) Assume that It is known that both “proposition p is true” & it is not known that “proposition p is true”. K(p & ~Kp)

(vii) It is known that “proposition p is true” & it is known that it is not known that “proposition p is true”. Kp & K~Kp [Logical Consequence of (vi) and (iv)]

(viii) It is known that “proposition p is true” & It is not known that “proposition p is true”. Kp & ~Kp [Logical Consequence of

(ix) It is not known that both “proposition p is true” & it is not known that “proposition p is true”.

(x) proven “proposition p is true” if and only if it is necessary that “proposition p is true”.

(xi) It is necessary that it is not known that both “proposition p is true” & it is not known that “proposition p is true”.

(xii) It is necessary that “proposition p is not true” if and only if it is not possible that “proposition p is true”.

(xiii) It is not possible that it is known that both “proposition p is true” & it is not known that “proposition p is true”.

(xiv) It is necessary that it is not known that both “proposition p is true” & it is not known that “proposition p is true”.